Primitive roots

[FONT='Times New Roman','serif']Let r be a primitive root of the odd prime p . Prove the following.[/FONT]
[FONT='Times New Roman','serif']If p is congruent to 3(mod 4), then -r has order (p-1)/2 modulo p.[/FONT]

[FONT='Times New Roman','serif']Let r be a primitive root of the odd prime p . Prove the following.[/FONT]
[FONT='Times New Roman','serif']If p is congruent to 3(mod 4), then -r has order (p-1)/2 modulo p.[/FONT]

.
Thus, it cannot be because the order of is .
Thus, .
This tells us, because .

Now we need to prove that if is order of then .
By the above result we see that , since is odd it follows that must be odd.
But then, .
Also, .
Using properties of primitive roots and orders it means .

Let r be a primitive root of the odd prime p. If p=1(mod 4), then -r is also a primitive root of p.

Let be the order of .
We claim that cannot be odd. Suppose that is odd.
Then .
Again which will force .
But that is a contradiction because so is even.
Since is even it mean and so the order must be the same as , i.e. .